MathDB

7

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 Team #7: Children and their Toy orderings

Source:

2/22/2013
There are are nn children and nn toys such that each child has a strict preference ordering on the toys. We want to distribute the toys: say a distribution AA dominates a distribution B≠A B \ne A if in AA, each child receives at least as preferable of a toy as in BB. Prove that if some distribution is not dominated by any other, then at least one child gets his/her favorite toy in that distribution.
2013 HMMT Algebra #7: Seven-Fold Summation

Source:

2/17/2013
Compute ∑a1=0∞∑a2=0∞⋯∑a7=0∞a1+a2+⋯+a73a1+a2+⋯+a7.\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.
HMMT
2013 HMMT Guts #7: Divisors of 15!

Source:

3/26/2013
Find the number of positive divisors dd of 15!=15⋅14⋅⋯⋅2⋅115!=15\cdot 14\cdot\cdots\cdot 2\cdot 1 such that gcd⁡(d,60)=5\gcd(d,60)=5.
HMMTnumber theoryprime factorization
2013 HMMT Geometry #7

Source:

3/3/2024
Let ABCABC be an obtuse triangle with circumcenter OO such that ∠ABC=15o\angle ABC = 15^o and ∠BAC>90o\angle BAC > 90^o. Suppose that AOAO meets BCBC at DD, and that OD2+OC⋅DC=OC2OD^2 + OC \cdot DC = OC^2. Find ∠C\angle C.
geometry